Excellent stuff! This paradox is a great specific example of Map-territory relation where the saying "All models are wrong (but some are useful)" originated. You even alluded to this idea towards the end. Essentially any conceptual/mathematical/logical model is not a perfect map of reality nor will it ever be. All we can do is try to continue minimizing the discrepancies between the model and our world. Additionally when we are building something with a specific model in mind we can ensure that we are aware of the discrepancies and ensure that those discrepancies are irrelevant to what we are building. A great entrance to the rabbit hole on these ideas here: https://en.m.wikipedia.org/wiki/Map%E2%80%93territory_relation
I think on model theory you need top be careful about "true". With regard to
"it’s like saying anything that’s first-order logically true of uncountably infinite sets like the real or complex numbers is true of the countables like the natural numbers"
I would respond:
True in a model is not true in the original axioms (unless true[provable] in all models). Some properties of uncountable sets, like the completeness of real numbers, can be expressed in first-order logic and are not true for countable sets like the rationals. It is not that notions are slippery. It is that provable in a model is not equivalent to provable in the axioms.
I think with incompleteness it is even the case that a statement provable from the axioms will be true in all models of those axioms, but not necessarily in every conceivable model but I'm not 100% sure. Just wanted to drop you a note.
I want to pose this question: was Putnam's dilemma ill-posed? One has no compulsion to take statements true[provable] of a model seriously. In analogy we have loads of Toy Models in physics. They are play things. Same in foundations of mathematics. Your Axioms are the foundation, not any particular model.
The fascinating thing, as your essay highlights, is the language problem. There are so many things in ordinary life that I think inherit the apparent paradox posed by Putnum. They'd not insignificant things.
It implies (to my mind) that the machine Ai systems cannot solve many of our social problems. They will always be dumb statistical inference algorithms, and so are subject to the language problems in Model theory. Since they are basically themselves machine models. This is something the AI tech bros fail to consider. Do0es anyone here agree or wish to amplify?
Just my opinion now: I think humans are not subject to the same limitations, since we are not merely biological machines. But we have severe limitations nonetheless, such as terrible memories, and slow computational ability. We more than make up with soulfulness or spiritual attributes, but the problems we can apply our minds to are in different categories. Thankfully. You never want technocracy running your politics. Whenever technocracy (currently being neoliberalism, previously the Soviets) is attempted it never ends well.
In your section 'Putnam's Paradoxical Playground', you say,
'It’s tempting to think that “something else” beyond our theories fixes the true interpretation'
I've seen this movie before! The 'something else beyond', reminds me of the role of 'hidden variables' in discussions of 'interpretations' of quantum mechanics. Which feeling of unease, if nothing else, provokes a stern examination of assumptions and inferences. Same here? Wonderful column, anyhow! Many thanks - as always!
I'm just now releasing this library of original works, I am not waiting any longer to finish the editing and layout completion of these papers.
Have several profound papers on logic including multilectic metalogic which is why I'm posting this here.
Please share this with anyone who might be interested.
Foundations of Hyperspace Physics
I am excited to invite you to explore our original revolutionary Library of Hyperspace Physics of over 330 papers, a collection of groundbreaking insights, theories, and frameworks that redefine our understanding of the universe.
From hypersymmetry and hypergravity to quantum coherence theory and the frontiers of cosmology to advanced technological designs this library offers a deep dive into the next evolution of physics and its transformative potential.
Start your journey into hyperspace discovery today and join a growing community of explorers reshaping the future of science.
Sometimes, it is difficult to imagine what lies beyond infinity. For example, is our world infinite or finite? Is there anything beyond its infinity? In religion, we often contemplate the eternal nature of life (whether good or bad). But does this mean we cannot imagine other forms of life beyond our temporal or physical world? To help visualize what might lie beyond infinity, we explore three examples.
Mapping the Axes3
When mapping a line to a circle, a straight line is treated as if it bends around to meet itself at infinity, forming a closed loop akin to a circle. This transformation can be thought of as "compactifying" the line, where infinity becomes just another point at the opposite side of zero. In this case, −∞ becomes next to +∞,
This visualization simplifies our understanding of infinity and allows us to imagine what might lie beyond it. In two-dimensional coordinates, for instance, we can visualize a ball. Anything beyond the surface of the ball exists beyond infinity, providing us with a way to perceive a world outside the coordinate axes.
Irrational Numbers
Irrational numbers do not have exact, finite values. Yet, we can measure them precisely in geometry. For example, consider the square root of 2 (√2), which is an irrational number. Its decimal expansion is non-terminating and non-repeating: approximately 1.414213... .
However, if we draw a right triangle with each leg of length 1, the hypotenuse will have a measurable, fixed value of √2. This provides another way to visualize an infinite, non-terminating concept as a concrete, measurable entity.
The Set Theory
In set theory, we distinguish between the members of a set and the set as a whole.
[a] ≠ a
For instance, consider the value of √2, which is approximately 1.414213.... If we place this value inside a set, we cannot fully reach its end because it is infinite in its decimal expansion. However, when we view the set from outside, we can conceptualize √2 as a fixed entity (e.g. length of the hypotenuse).
Similarly, take an infinite geometric series such as: 1+ ½ + ¼ + 1/8 + … . From within the set, the series appears infinite, but when observed from outside, it has a fixed, exact value of 2. This transition from infinite to finite highlights how stepping outside a given realm allows us to comprehend the infinite.
Conclusion
To visualize or comprehend an infinite entity, we must step outside its realm. As long as we remain within that realm, reaching infinity is impossible by definition. However, by defining a new perspective or realm, we can easily understand and conceptualize what exists beyond infinity.
And I'm not sure how you can have an uncountable universe without its corresponding yang, as consciousness itself demonstrates when considered as the fundamental substrate of our reality. Math was never my strong suit (not counting Geometry) and can't read a note of music but, like Cat Stevens, have been writing (and playing) songs since the age of 10 almost 60 years ago :)
It must be acknowledged that Cantor's diagonal proof already assumes that actual infinity can be taken; there is no prior demonstration that you can proceed as is done in that proof. It has, at most in mathematics, a subjective nuance, but we favor actual infinity because it brings very interesting and valuable mathematics. However, just like Platonic objects or any theorem, it does not exist in physical reality; the Pythagorean theorem does not hold rigorously in nature, nor does any other theorem.
To conclude, and to reinforce the accurate conclusion you reach at the end of the article, it must be understood that mathematics is not the language of nature, but it helps us reach where intuition cannot.
It’s refreshing to see paradoxes other than Gödel’s famous theorems. Another (simpler one) is Tarski's undefinability theorem.
I am a bit surprised that you say it isn’t relativism though. To me it really sounds like a case of countability being relative. The uncoutability of the Real remains true as a sentence inside the first order model even if that model is countable. This is because the countability of the model is “external” and refers to the signature of that model from the “outside”.
Inside the model, the sentence saying that the Reals are uncountable remains True which is a clear case of relativism of infinity, wouldn’t you agree?
CJ: “The universe is something you engage with. Something you try to make sense of using the tools you have.”
Great conclusion and a great quote.
Totally. @curt In the spirit of social semantics.. how about “in”, “with/in”, or “with…and…in”?
Excellent stuff! This paradox is a great specific example of Map-territory relation where the saying "All models are wrong (but some are useful)" originated. You even alluded to this idea towards the end. Essentially any conceptual/mathematical/logical model is not a perfect map of reality nor will it ever be. All we can do is try to continue minimizing the discrepancies between the model and our world. Additionally when we are building something with a specific model in mind we can ensure that we are aware of the discrepancies and ensure that those discrepancies are irrelevant to what we are building. A great entrance to the rabbit hole on these ideas here: https://en.m.wikipedia.org/wiki/Map%E2%80%93territory_relation
I think on model theory you need top be careful about "true". With regard to
"it’s like saying anything that’s first-order logically true of uncountably infinite sets like the real or complex numbers is true of the countables like the natural numbers"
I would respond:
True in a model is not true in the original axioms (unless true[provable] in all models). Some properties of uncountable sets, like the completeness of real numbers, can be expressed in first-order logic and are not true for countable sets like the rationals. It is not that notions are slippery. It is that provable in a model is not equivalent to provable in the axioms.
I think with incompleteness it is even the case that a statement provable from the axioms will be true in all models of those axioms, but not necessarily in every conceivable model but I'm not 100% sure. Just wanted to drop you a note.
Kant’s transcendental idealism reigns undefeated as always
I want to pose this question: was Putnam's dilemma ill-posed? One has no compulsion to take statements true[provable] of a model seriously. In analogy we have loads of Toy Models in physics. They are play things. Same in foundations of mathematics. Your Axioms are the foundation, not any particular model.
The fascinating thing, as your essay highlights, is the language problem. There are so many things in ordinary life that I think inherit the apparent paradox posed by Putnum. They'd not insignificant things.
It implies (to my mind) that the machine Ai systems cannot solve many of our social problems. They will always be dumb statistical inference algorithms, and so are subject to the language problems in Model theory. Since they are basically themselves machine models. This is something the AI tech bros fail to consider. Do0es anyone here agree or wish to amplify?
Just my opinion now: I think humans are not subject to the same limitations, since we are not merely biological machines. But we have severe limitations nonetheless, such as terrible memories, and slow computational ability. We more than make up with soulfulness or spiritual attributes, but the problems we can apply our minds to are in different categories. Thankfully. You never want technocracy running your politics. Whenever technocracy (currently being neoliberalism, previously the Soviets) is attempted it never ends well.
In your section 'Putnam's Paradoxical Playground', you say,
'It’s tempting to think that “something else” beyond our theories fixes the true interpretation'
I've seen this movie before! The 'something else beyond', reminds me of the role of 'hidden variables' in discussions of 'interpretations' of quantum mechanics. Which feeling of unease, if nothing else, provokes a stern examination of assumptions and inferences. Same here? Wonderful column, anyhow! Many thanks - as always!
Hello,
I'm just now releasing this library of original works, I am not waiting any longer to finish the editing and layout completion of these papers.
Have several profound papers on logic including multilectic metalogic which is why I'm posting this here.
Please share this with anyone who might be interested.
Foundations of Hyperspace Physics
I am excited to invite you to explore our original revolutionary Library of Hyperspace Physics of over 330 papers, a collection of groundbreaking insights, theories, and frameworks that redefine our understanding of the universe.
From hypersymmetry and hypergravity to quantum coherence theory and the frontiers of cosmology to advanced technological designs this library offers a deep dive into the next evolution of physics and its transformative potential.
Start your journey into hyperspace discovery today and join a growing community of explorers reshaping the future of science.
Invitation Link:
https://app.box.com/s/jyd1w7d98c0en4ucegpmje6l3ddg67w9
Best regards,
Philip Randolph Adam Lilien
hhhcr-toe@hyperspacefoundations.com
Sometimes, it is difficult to imagine what lies beyond infinity. For example, is our world infinite or finite? Is there anything beyond its infinity? In religion, we often contemplate the eternal nature of life (whether good or bad). But does this mean we cannot imagine other forms of life beyond our temporal or physical world? To help visualize what might lie beyond infinity, we explore three examples.
Mapping the Axes3
When mapping a line to a circle, a straight line is treated as if it bends around to meet itself at infinity, forming a closed loop akin to a circle. This transformation can be thought of as "compactifying" the line, where infinity becomes just another point at the opposite side of zero. In this case, −∞ becomes next to +∞,
This visualization simplifies our understanding of infinity and allows us to imagine what might lie beyond it. In two-dimensional coordinates, for instance, we can visualize a ball. Anything beyond the surface of the ball exists beyond infinity, providing us with a way to perceive a world outside the coordinate axes.
Irrational Numbers
Irrational numbers do not have exact, finite values. Yet, we can measure them precisely in geometry. For example, consider the square root of 2 (√2), which is an irrational number. Its decimal expansion is non-terminating and non-repeating: approximately 1.414213... .
However, if we draw a right triangle with each leg of length 1, the hypotenuse will have a measurable, fixed value of √2. This provides another way to visualize an infinite, non-terminating concept as a concrete, measurable entity.
The Set Theory
In set theory, we distinguish between the members of a set and the set as a whole.
[a] ≠ a
For instance, consider the value of √2, which is approximately 1.414213.... If we place this value inside a set, we cannot fully reach its end because it is infinite in its decimal expansion. However, when we view the set from outside, we can conceptualize √2 as a fixed entity (e.g. length of the hypotenuse).
Similarly, take an infinite geometric series such as: 1+ ½ + ¼ + 1/8 + … . From within the set, the series appears infinite, but when observed from outside, it has a fixed, exact value of 2. This transition from infinite to finite highlights how stepping outside a given realm allows us to comprehend the infinite.
Conclusion
To visualize or comprehend an infinite entity, we must step outside its realm. As long as we remain within that realm, reaching infinity is impossible by definition. However, by defining a new perspective or realm, we can easily understand and conceptualize what exists beyond infinity.
The map is not the terrain ;)
And I'm not sure how you can have an uncountable universe without its corresponding yang, as consciousness itself demonstrates when considered as the fundamental substrate of our reality. Math was never my strong suit (not counting Geometry) and can't read a note of music but, like Cat Stevens, have been writing (and playing) songs since the age of 10 almost 60 years ago :)
Thanks, Curt, for such a provocative post!
It must be acknowledged that Cantor's diagonal proof already assumes that actual infinity can be taken; there is no prior demonstration that you can proceed as is done in that proof. It has, at most in mathematics, a subjective nuance, but we favor actual infinity because it brings very interesting and valuable mathematics. However, just like Platonic objects or any theorem, it does not exist in physical reality; the Pythagorean theorem does not hold rigorously in nature, nor does any other theorem.
To conclude, and to reinforce the accurate conclusion you reach at the end of the article, it must be understood that mathematics is not the language of nature, but it helps us reach where intuition cannot.
It’s refreshing to see paradoxes other than Gödel’s famous theorems. Another (simpler one) is Tarski's undefinability theorem.
I am a bit surprised that you say it isn’t relativism though. To me it really sounds like a case of countability being relative. The uncoutability of the Real remains true as a sentence inside the first order model even if that model is countable. This is because the countability of the model is “external” and refers to the signature of that model from the “outside”.
Inside the model, the sentence saying that the Reals are uncountable remains True which is a clear case of relativism of infinity, wouldn’t you agree?